I'm running Firefox 3.0.4 on two machines: a desktop X86-64 running OpenSuSE 11.0 and a laptop running Kubuntu Intrepid 8.10. On the desktop, Firefox frequently simply quits, with no relevant messages in /var/log/messages. On the laptop I also get the same behavior but much less often. I don't even know where to find a log that might indicate the cause of the problem. I've had the same problem with other versions of Firefox in the 3.0.x series. Any advice?

Firefox is unlikely to log things to /var/log/messages; it's much more likely to go into the X logs, usually /var/log/XOrg.0.log or similar.

Most crashes in Firefox can be tracked down to a misbehaving plugin or extension.

You might start simply by comparing the installed plugins or extensions on the two machines. Also, uninstall or disable any plugins or extensions that you're not using, and update the rest to the latest versions (the update button in the Tools->Add-ons dialog).

There are a few tools for tracking down misbehaving plugins; a cursory search yields this https://addons.mozilla.org/en-US/firefox/addon/7055 (although it's not available on Linux, I'm sure you could find similar).

Also, make sure that other programs don't exhibit weird symptoms; there's an outside chance that it could be a hardware fault; you might try running a memory tester program overnight.